1.
30 (number)
–
30 is the natural number following 29 and preceding 31. Thirty is the sum of the first four squares, which makes it a square pyramidal number and it is a primorial and is the smallest Giuga number. 30 is the smallest sphenic number, and the smallest of the form 2 ×3 × r,30 has an aliquot sum of 42, the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises Thirty has but one number for which it is the aliquot sum, adding up some subsets of its divisors gives 30, hence 30 is a semiperfect number. 30 is the largest number such that all smaller than itself. A polygon with thirty sides is called a triacontagon, the icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a graph with 30 vertices. The atomic number of zinc is 30 Messier object M30, a magnitude 8, the duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series began on June 19,1803 BC. The duration of Saros series 30 was 1316.2 years, Thirty is, Used to indicate the end of a newspaper story, a copy editors typographical notation. S. Judas Iscariot betrayed Jesus for 30 pieces of silver, one of the rallying-cries of the 1960s student/youth protest movement was the slogan, Dont trust anyone over thirty. In Franz Kafkas novel The Trial Joseph wakes up on the morning of his birthday to find himself under arrest for an unspecified crime. After making many attempts to find the nature of the crime or the name of his accuser. The number of uprights that formed the Sarsen Circle at Stonehenge, western Christianitys most prolific 20th century essayist, F. W. Also in that essay Boreham writes It was said of Keats, in tennis, the number 30 represents the second point gained in a game. Under NCAA rules for basketball, the offensive team has 30 seconds to attempt a shot. As of 2012, three of the four major leagues in the United States and Canada have 30 teams each. The California Angels baseball team retired the number in honor of its most notable wearer, Nolan Ryan, the San Francisco Giants extended the same honor to Orlando Cepeda
30 (number)
–
For other uses, see
The Thirty.
2.
19 (number)
–
19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
19 (number)
–
A 19x19
Go board
19 (number)
–
19 is a
centered triangular number
3.
20 (number)
–
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score,20 is a tetrahedral number as 1,4,10,20. 20 is the basis for vigesimal number systems,20 is the third composite number comprising the product of a squared prime and a prime, and also the second member of the q family in this form. 20 has a sum of 22. Accordingly,20 is the abundant number and demonstrates an 8-member aliquot sequence. 20 is the smallest primitive abundant number,20 is the 4th composite number in the 7-aliquot tree. Two numbers have 20 as their sum, the discrete semiprime 34. Only 2 other square primes are abundant 12 and 18,20 can be written as the sum of three Fibonacci numbers uniquely, i. e.20 =13 +5 +2. The product of the number of divisors and the number of divisors of 20 is exactly 20. 20 is the number of required to optimally solve a Rubiks Cube in the worst case. 20 is the number with more than one digit that can be written from base 2 to base 20 using only the digits 0 to 9. The third magic number in physics, the IAU shower number for Coma Berenicids. The number of amino acids that are encoded by the standard genetic code. In some countries, the number 20 is used as an index in measuring visual acuity, 20/20 indicates normal vision at 20 feet, although it is commonly used to mean perfect vision. When someone is able to see only after an event how things turned out, the Baltimore Orioles and Cincinnati Reds, both for Hall of Famer Frank Robinson. The Kansas City Royals, for Frank White, the Los Angeles Dodgers, for Hall of Famer Don Sutton. The Philadelphia Phillies, for Hall of Famer Mike Schmidt, the Pittsburgh Pirates, for Hall of Famer Pie Traynor. The St. Louis Cardinals, for Hall of Famer Lou Brock, the San Francisco Giants, for Hall of Famer Monte Irvin, who played for the team when it was the New York Giants
20 (number)
–
An
icosahedron has 20
faces
4.
21 (number)
–
21 is the natural number following 20 and preceding 22. In a 24-hour clock, the twenty-first hour is in conventional language called nine or nine oclock,21 is, the fifth discrete semiprime and the second in the family. With 22 it forms the second discrete semiprime pair, a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. A composite number, its divisors being 1,3 and 7. The sum of the first six numbers, making it a triangular number. The sum of the sum of the divisors of the first 5 positive integers, the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number. The smallest natural number that is not close to a power of 2, 2n,21 has an aliquot sum of 11 though it is the second composite number found in the 11-aliquot tree with the abundant square prime 18 being the first such member. Twenty-one is the first number to be the sum of three numbers 18,51,91. 21 appears in the Padovan sequence, preceded by the terms 9,12,16, in several countries 21 is the age of majority. In most US states,21 is the drinking age, however, in Puerto Rico and U. S. Virgin Island, the drinking age is 18. In Hawaii and New York,21 is the age that one person may purchase cigarettes. In some countries it is the voting age, in the United States,21 is the age at which one can purchase multiple tickets to an R-rated film without providing Identifications. It is also the age to one under the age of 17 as their parent or adult guardian for an R-rated movie. In some states,21 is the age, persons may gamble or enter casinos. In 2011, Adele named her second studio album 21, because of her age at the time, the Milwaukee Braves, for Hall of Famer Warren Spahn, the number continues to be honored by the team in its current home of Atlanta. The Pittsburgh Pirates, for Hall of Famer Roberto Clemente, following his death in a crash while attempting to deliver humanitarian aid to victims of an earthquake in Nicaragua. In the NBA, The Atlanta Hawks, for Hall of Famer Dominique Wilkins, the Boston Celtics, for Hall of Famer Bill Sharman. The Detroit Pistons, for Hall of Famer Dave Bing, the Sacramento Kings, for Vlade Divac
21 (number)
–
Number 21 on the road bicycle of
Ellen van Dijk at the
Ronde van Drenthe.
21 (number)
–
Building called "21" in
Zlín,
Czech Republic.
21 (number)
–
Detail of the building entrance
5.
24 (number)
–
24 is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta, and for 10−24 yocto and these numbers are the largest and smallest number to receive an SI prefix to date. In a 24-hour clock, the hour is in conventional language called twelve or twelve oclock. 24 is the factorial of 4 and a number, being the first number of the form 23q. It follows that 24 is the number of ways to order 4 distinct items and it is the smallest number with exactly eight divisors,1,2,3,4,6,8,12, and 24. It is a composite number, having more divisors than any smaller number. 24 is a number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. Subtracting 1 from any of its divisors yields a number,24 is the largest number with this property. 24 has a sum of 36 and the aliquot sequence. It is therefore the lowest abundant number whose aliquot sum is itself abundant, the aliquot sum of only one number,529 =232, is 24. There are 10 solutions to the equation φ =24, namely 35,39,45,52,56,70,72,78,84 and 90 and this is more than any integer below 24, making 24 a highly totient number. 24 is the sum of the prime twins 11 and 13, the product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two numbers, one of which is a multiple of four, and there must be a multiple of three. The tesseract has 24 two-dimensional faces,24 is the only nontrivial solution to the cannonball problem, that is,12 +22 +32 + … +242 is a perfect square. In 24 dimensions there are 24 even positive definite unimodular lattices, the Leech lattice is closely related to the equally nice length-24 binary Golay code and the Steiner system S and the Mathieu group M24. The modular discriminant Δ is proportional to the 24th power of the Dedekind eta function η, Δ = 12η24, the Barnes-Wall lattice contains 24 lattices. 24 is the number whose divisors — namely 1,2,3,4,6,8,12,24 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group × = is isomorphic to the additive group 3 and this fact plays a role in monstrous moonshine
24 (number)
–
Astronomical clock in Prague
6.
25 (number)
–
25 is the natural number following 24 and preceding 26. It is a number, being 52 =5 ×5. It is one of two numbers whose square and higher powers of the number also ends in the same last two digits, e. g.252 =625, the other is 76. It is the smallest square that is also a sum of two squares,25 =32 +42, hence it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the odd natural numbers 1,3,5,7 and 9. 25 is an octagonal number, a centered square number. 25 percent is equal to 1/4,25 has an aliquot sum of 6 and number 6 is the first number to have an aliquot sequence that does not culminate in 0 through a prime. 25 is the sum of three integers,95,119, and 143. Twenty-five is the second member of the 6-aliquot tree. It is the smallest base 10 Friedman number as it can be expressed by its own digits,52 and it is also a Cullen number. 25 is the smallest pseudoprime satisfying the congruence 7n =7 mod n.25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate. Within base 10 one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00,25,50 or 75. 25 and 49 are the perfect squares in the following list,13,25,37,49,511,613,715,817,919,1021,1123,1225,1327,1429. The formula in this list can be described as 10nZ + where n depends on the number of digits in Z, in base 30,25 is a 1-automorphic number, and in base 10 a 2-automorphic number. The percent DNA overlap of a half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin, in Ezekiels vision of a new temple, The number twenty-five is of cardinal importance in Ezekiels Temple Vision. In The Book of Revelations New International Version, Surrounding the throne were twenty-four other thrones and they were dressed in white and had crowns of gold on their heads. In Islam, there are 25 prophets mentioned in the Quran, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expand their roster to 40 players. The size of the roster on a Nippon Professional Baseball team for a particular game
25 (number)
–
25 is a square
7.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
8.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
9.
40 (number)
–
Despite being related to the word four, the modern spelling of 40 is forty. The archaic form fourty is now considered a misspelling, the modern spelling possibly reflects a pronunciation change due to the horse–hoarse merger. Forty is a number, an octagonal number, and as the sum of the first four pentagonal numbers. Adding up some subsets of its divisors gives 40, hence 40 is a semiperfect number, given 40, the Mertens function returns 0. 40 is the smallest number n with exactly 9 solutions to the equation φ = n, Forty is the number of n-queens problem solutions for n =7. Since 402 +1 =1601 is prime,40 is a Størmer number,40 is a repdigit in base 3 and a Harshad number in base 10. Negative forty is the temperature at which the Fahrenheit and Celsius scales correspond. It is referred to as either minus forty or forty below, the planet Venus forms a pentagram in the night sky every eight years with it returning to its original point every 40 years with a 40-day regression. The duration of Saros series 40 was 1280.1 years, lunar eclipse series which began on -1387 February 12 and ended on -71 April 12. The duration of Saros series 40 was 1316.2 years, the number 40 is used in Jewish, Christian, Islamic, and other Middle Eastern traditions to represent a large, approximate number, similar to umpteen. In the Hebrew Bible, forty is often used for periods, forty days or forty years. Rain fell for forty days and forty nights during the Flood, spies explored the land of Israel for forty days. The Hebrew people lived in the Sinai desert for forty years and this period of years represents the time it takes for a new generation to arise. Moses life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out, several Jewish leaders and kings are said to have ruled for forty years, that is, a generation. Examples include Eli, Saul, David, and Solomon, goliath challenged the Israelites twice a day for forty days before David defeated him. He went up on the day of Tammuz to beg forgiveness for the peoples sin. He went up on the first day of Elul and came down on the day of Tishrei. A mikvah consists of 40 seah of water 40 lashes is one of the punishments meted out by the Sanhedrin, One of the prerequisites for a man to study Kabbalah is that he is forty years old
40 (number)
–
The number on the logo for the American-Japanese hard rock band Crush 40.
10.
60 (number)
–
60 is the natural number following 59 and preceding 61. Being three times 20, it is called three score in older literature. It is a number, with divisors 1,2,3,4,5,6,10,12,15,20,30. Because it is the sum of its divisors, it is a unitary perfect number. Being ten times a number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6 and it is the smallest number with exactly 12 divisors. It is the sum of a pair of twin primes and the sum of four consecutive primes and it is adjacent to two primes. It is the smallest number that is the sum of two odd primes in six ways, the smallest non-solvable group has order 60. There are four Archimedean solids with 60 vertices, the icosahedron, the rhombicosidodecahedron, the snub dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs, there are also two Archimedean solids with 60 edges, the snub cube and the icosidodecahedron. The skeleton of the forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares, in geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, a number system with base 60 is called sexagesimal. It is the smallest positive integer that is written only the smallest. The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule and this ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 is an isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz
60 (number)
–
There are 60 seconds in a minute, and 60 minutes in an hour
60 (number)
–
The
icosidodecahedron has 60 edges, all equivalent.
11.
80 (number)
–
80 is the natural number following 79 and preceding 81. 80 is, the sum of Eulers totient function φ over the first sixteen integers, a semiperfect number, since adding up some subsets of its divisors gives 80. Palindromic in bases 3,6,9,15,19 and 39, a repdigit in bases 3,9,15,19 and 39. A Harshad number in bases 2,3,4,5,6,7,9,10,11,13,15 and 16 The Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes. Every solvable configuration of the Fifteen puzzle can be solved in no more than 80 single-tile moves, the atomic number of mercury According to Exodus 7,7, Moses was 80 years old when he initially spoke to Pharaoh on behalf of his people. Today,80 years of age is the age limit for cardinals to vote in papal elections. Jerry Rice wore the number 80 for the majority of his NFL career
80 (number)
–
Element 80: Mercury (Hg)
12.
90 (number)
–
90 is the natural number preceded by 89 and followed by 91. In English speech, the numbers 90 and 19 are often confused, when carefully enunciated, they differ in which syllable is stressed,19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in such as 1999, and when contrasting numbers in the teens and when counting, such as 17,18,19. 90 is, a perfect number because it is the sum of its unitary divisors. A semiperfect number because it is equal to the sum of a subset of its divisors, a Perrin number, preceded in the sequence by 39,51,68. Palindromic and a repdigit in bases 14,17,29, a Harshad number since 90 is divisible by the sum of its base 10 digits. In normal space, the angles of a rectangle measure 90 degrees each. Also, in a triangle, the angle opposing the hypotenuse measures 90 degrees. Thus, an angle measuring 90 degrees is called a right angle, ninety is, the atomic number of thorium, an actinide. As an atomic weight,90 identifies an isotope of strontium, the latitude in degrees of the North and the South geographical poles. NFL, New York Jets Dennis Byrds #90 is retired +90 is the code for international direct dial phone calls to Turkey,90 is the code for the French département Belfort
90 (number)
–
Interstate 90 is a freeway that runs from
Washington to
Massachusetts.
13.
100 (number)
–
100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the hundred or five score in order to differentiate the English. The standard SI prefix for a hundred is hecto-,100 is the basis of percentages, with 100% being a full amount. 100 is the sum of the first nine prime numbers, as well as the sum of pairs of prime numbers e. g.3 +97,11 +89,17 +83,29 +71,41 +59. 100 is the sum of the cubes of the first four integers and this is related by Nicomachuss theorem to the fact that 100 also equals the square of the sum of the first four integers,100 =102 =2. 26 +62 =100, thus 100 is a Leyland number and it is divisible by the number of primes below it,25 in this case. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient and it can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, there are exactly 100 prime numbers whose digits are in strictly ascending order. 100 is the smallest number whose common logarithm is a prime number,100 senators are in the U. S One hundred is the atomic number of fermium, an actinide. On the Celsius scale,100 degrees is the temperature of pure water at sea level. The Kármán line lies at an altitude of 100 kilometres above the Earths sea level and is used to define the boundary between Earths atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of Rosh Hashana, a religious Jew is expected to utter at least 100 blessings daily. In Hindu Religion - Mythology Book Mahabharata - Dhritarashtra had 100 sons known as kauravas, the United States Senate has 100 Senators. Most of the currencies are divided into 100 subunits, for example, one euro is one hundred cents. The 100 Euro banknotes feature a picture of a Rococo gateway on the obverse, the U. S. hundred-dollar bill has Benjamin Franklins portrait, the Benjamin is the largest U. S. bill in print. American savings bonds of $100 have Thomas Jeffersons portrait, while American $100 treasury bonds have Andrew Jacksons portrait, One hundred is also, The number of years in a century. The number of pounds in an American short hundredweight, in Greece, India, Israel and Nepal,100 is the police telephone number. In Belgium,100 is the ambulance and firefighter telephone number, in United Kingdom,100 is the operator telephone number
100 (number)
–
The
U.S. hundred-dollar bill, Series 2009.
14.
Factorization
–
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
Factorization
–
A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.
15.
Prime number
–
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
–
The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
16.
Divisor
–
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
Divisor
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The divisors of 10 illustrated with
Cuisenaire rods: 1, 2, 5, and 10
17.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
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Numeral systems
Greek numerals
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A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
18.
Roman numerals
–
The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
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Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
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Numeral systems
Roman numerals
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A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
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An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
19.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
–
Gottfried Leibniz
Binary number
–
George Boole
20.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
Ternary numeral system
–
Numeral systems
21.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
Quaternary numeral system
–
Numeral systems
22.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
Quinary
–
Numeral systems
23.
Senary
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Senary
–
Numeral systems
Senary
–
34 senary = 22 decimal, in senary finger counting
Senary
24.
Octal
–
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
Octal
–
Numeral systems
25.
Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
–
Numeral systems
Duodecimal
–
A duodecimal multiplication table
26.
Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
Hexadecimal
–
Numeral systems
Hexadecimal
–
Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
–
Hexadecimal finger-counting scheme.
27.
Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
Vigesimal
–
Numeral systems
Vigesimal
–
The
Maya numerals are a base-20 system.
28.
Base 36
–
The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
Base 36
–
Numeral systems
Base 36
–
34 senary = 22 decimal, in senary finger counting
Base 36
29.
Natural number
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural number
–
The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
30.
31 (number)
–
31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages
31 (number)
–
31 is a
centered pentagonal number
31.
Pell number
–
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers
Pell number
–
Rational approximations to regular
octagons, with coordinates derived from the Pell numbers.
32.
Eisenstein prime
–
In mathematics, an Eisenstein prime is an Eisenstein integer z = a + b ω that is irreducible in the ring-theoretic sense, its only Eisenstein divisors are the units, a + bω itself and its associates. The associates and the conjugate of any Eisenstein prime are also prime. It follows that the absolute value squared of every Eisenstein prime is a prime or the square of a natural prime. The first few Eisenstein primes that equal a natural prime 3n −1 are,2,5,11,17,23,29,41,47,53,59,71,83,89,101. Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes, some non-real Eisenstein primes are 2 + ω,3 + ω,4 + ω,5 + 2ω,6 + ω,7 + ω,7 + 3ω. Up to conjugacy and unit multiples, the primes listed above, as of March 2017, the largest known Eisenstein prime is the seventh largest known prime 10223 ×231172165 +1, discovered by Péter Szabolcs and PrimeGrid. All larger known primes are Mersenne primes, discovered by GIMPS, real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes are congruent to 1 mod 3, thus no Mersenne prime is an Eisenstein prime
Eisenstein prime
–
Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3 n − 1. All others have an absolute value squared equal to a natural prime.
33.
Markov number
–
The first few Markov numbers are 1,2,5,13,29,34,89,169,194,233,433,610,985,1325. Appearing as coordinates of the Markov triples, etc, there are infinitely many Markov numbers and Markov triples. There are two ways to obtain a new Markov triple from an old one. First, one may permute the 3 numbers x, y, z, second, if is a Markov triple then by Vieta jumping so is. Applying this operation twice returns the same triple one started with, joining each normalized Markov triple to the 1,2, or 3 normalized triples one can obtain from this gives a graph starting from as in the diagram. This graph is connected, in other words every Markov triple can be connected to by a sequence of these operations. If we start, as an example, with we get its three neighbors, and in the Markov tree if x is set to 1,5 and 13, respectively. For instance, starting with and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers, starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers. All the Markov numbers on the adjacent to 2s region are odd-indexed Pell numbers. Thus, there are infinitely many Markov triples of the form, likewise, there are infinitely many Markov triples of the form, where Px is the xth Pell number. Aside from the two smallest singular triples and, every Markov triple consists of three distinct integers, odd Markov numbers are 1 more than multiples of 4, while even Markov numbers are 2 more than multiples of 32. In his 1982 paper, Don Zagier conjectured that the nth Markov number is given by m n =13 e C n + o with C =2.3523414972 …. Moreover, he pointed out that x 2 + y 2 + z 2 =3 x y z +4 /9, the conjecture was proved by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry. The nth Lagrange number can be calculated from the nth Markov number with the formula L n =9 −4 m n 2, the Markov numbers are sums of pairs of squares. If X⋅Y⋅Z =1 then Tr = Tr, so more symmetrically if X, Y, and Z are in SL2 with X⋅Y⋅Z =1, cambridge Tracts in Mathematics and Mathematical Physics. Markov spectrum problem, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Markoff, A. Sur les formes quadratiques binaires indéfinies
Markov number
–
The first levels of the Markov number tree
34.
Bishnois
–
Bishnoi is a religious group found in the Western Thar Desert and northern states of India. They follow a set of 29 principles given by Guru Jambheshwar, Jambheshwar, who lived in the 15th century, said that trees and wildlife should be protected, prophesying that harming the environment means harming oneself. Bishnoism was founded by Guru Jambheshwar of Bikaner, who was born in 1499 and he announced a set of 29 tenets. The name Bishnoi was derived from Vishnu. These were contained in a document written in the Nagri script called Shabdwani, of his 29 tenets, ten are directed towards personal hygiene and maintaining good basic health, seven for healthy social behaviour, and four tenets to the worship of God. Eight tenets have been prescribed to preserve bio-diversity and encourage good animal husbandry and these include a ban on killing animals and felling green trees, and providing protection to all life forms. The community is also directed to see that the firewood they use is devoid of small insects, wearing blue clothes is prohibited because the dye for colouring them is obtained by cutting a large quantity of shrubs. Bishnois are found in Rajasthan, Haryana, Punjab and Uttar Pradesh and they do not allow intercaste marriage. The most prominent places of pilgrimage of the Bishnois is situated at village called Mukam in a temple in Nokha Tehsil, Bikaner District, Rajasthan. The Bishnoi narrate the story of Amrita Devi, a Bishnoi woman who, along more than 363 other Bishnois. Nearly two centuries ago, Maharajah Abhay Singh of Jodhpur required wood for the construction of his new palace, so the king sent his soldiers to cut trees in the nearby region of Khejarli, where the village is filled with the large number of trees. But when Amrita Devi and local villagers came to know about it, the feudal party told her that if she wanted the trees to be spared, she would have to give them money as a bribe. She refused to acknowledge this demand and told them that she would consider it as an act of insult to her religious faith and this is still remembered as the great Khejarli sacrifice. Some Bishnois who were killed protecting the trees were buried in Khejerli village near Jodhpur, every year, in September, the Bishnois assemble there to commemorate the extreme sacrifice made by their people to preserve their faith and religion. The 29 tenets of Bishnoism state, Observe 30 days state of ritual impurity after childs birth and keep mother, Observe 5 days segregation while a woman is in her menses. Take bath daily in the morning, obey the ideal rules of life, Modesty, Patience or satisfactions, cleanliness. Eulogise God, Vishnu, in evening hours Perform Yajna with the feelings of welfare devotion, use filtered water, milk and cleaned firewood. Speak pure words in all sincerity, dont steal or keep any intention to do it. Worship and recite Lord Vishnu in adoration Have merciful on all living beings, do not cut green trees, save the environment
Bishnois
–
Guru Jambheshwar also known Jambhaji, is the spiritual leader of Bishnoi sect.
35.
Guru Jambheshwar
–
Shree Guru Jambeshwar Bhagwan, also known as Jambho ji, was the founder of the Bishnoi sect. He preached the worship of Hari and he taught that God is a divine power that is everywhere. He also taught to protect plants and animals as they are important in order to peacefully coexist with nature and he was the only child of his parents i. e. father, Lohat ji Panwar and mother, Hansa Devi. For the first seven years of his life, Jambho ji was considered silent and he spent 27 years of his life as a cow herder like Lord Krishna, with whom he shared a birthday. At an age of 34, Jambho ji founded the Bishnoi sect and his teachings were in the poetic form known as Shabadwani. Although he preached for the next 51 years, travelling across the country, only 120 Shabads, even these 120 shabads are a source of great wisdom and are sufficient for an individual to understand and follow his path. Bishnoism, as mention earlier revolves around 29 commandments, out of these 29 commandments,8 prescribe to preserve biodiversity and encourage good animal husbandry. Seven commandments provide directions for social behaviour. Ten commandments are directed towards personal hygiene and maintaining good health. The other four commandments provide guidelines for worshipping God daily and he saw the need for environmental protection and weaved his principals into religious commandments so that people can internalise those principals easily. The sect was founded by Shree Guru Jambeshwar Bhagwan after wars between Muslim invaders and local Hindus and he had laid down 29 principles to be followed by the sect. Bish means twenty and noi means nine, killing animals and felling trees were banned. The Prosopis cineraria tree, is also considered to be sacred by the Bishnois. His old father shri Lohat Ji Panwar was very sad when commented by a farmer for not having children up to 50 years of age, then he started tapsya and was blessed by a yogi for a son who will be different from others. Same time his wife Hansa Devi was also blessed by the same yogi for a son, considering the blessings of yogi, miracle powers of Jambo Ji, Guru Ji’s shabads, Jambheswar ji is popularly considered Vishnu sawrup. The young Jambh Raj did not drink milk from his mothers breast and he told first shabad to a Brahmin called to cure his dumbness. The young Jambh dev was simple but a genius, kind, liked loneliness and he did not marry and used to graze cows. At the age of 34 he left his home and belongings and he was very keen in social welfare and helping others. In year 1485 there was a worst drought in western Rajasthan area, the kind-hearted Jambha ji was sad to see people’s pain
Guru Jambheshwar
–
Shree Guru Jambeshwar Bhagwan
Guru Jambheshwar
–
The marble tomb at Jambhoji Dham at Mukam in Nagaur district, Rajasthan
Guru Jambheshwar
–
Khejarli Massacre in 1730 in which 363 men, women and children of Bishnoi community laid down their lives to protect trees from cutting.
36.
Atomic number
–
The atomic number or proton number of a chemical element is the number of protons found in the nucleus of an atom of that element. It is identical to the number of the nucleus. The atomic number identifies a chemical element. In an uncharged atom, the number is also equal to the number of electrons. The atomic number Z, should not be confused with the mass number A and this number of neutrons, N, completes the weight, A = Z + N. Atoms with the atomic number Z but different neutron numbers N. Historically, it was these atomic weights of elements that were the quantities measurable by chemists in the 19th century. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge, loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order. Dmitri Mendeleev claimed that he arranged his first periodic tables in order of atomic weight, however, in consideration of the elements observed chemical properties, he changed the order slightly and placed tellurium ahead of iodine. This placement is consistent with the practice of ordering the elements by proton number, Z. A simple numbering based on periodic table position was never entirely satisfactory and this central charge would thus be approximately half the atomic weight. This proved eventually to be the case, the experimental position improved dramatically after research by Henry Moseley in 1913. To do this, Moseley measured the wavelengths of the innermost photon transitions produced by the elements from aluminum to gold used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons increased from one target to the next in an arithmetic progression and this led to the conclusion that the atomic number does closely correspond to the calculated electric charge of the nucleus, i. e. the element number Z. Among other things, Moseley demonstrated that the series must have 15 members—no fewer. After Moseleys death in 1915, the numbers of all known elements from hydrogen to uranium were examined by his method. There were seven elements which were not found and therefore identified as still undiscovered, from 1918 to 1947, all seven of these missing elements were discovered. By this time the first four transuranium elements had also been discovered, in 1915 the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood
Atomic number
–
An explanation of the superscripts and subscripts seen in atomic number notation. Atomic number is the number of protons, and therefore also the total positive charge, in the atomic nucleus.
Atomic number
–
Russian chemist Dmitri Mendeleev created a periodic table of the elements that ordered them numerically by atomic weight, yet occasionally used chemical properties in contradiction to weight.
Atomic number
–
Niels Bohr's 1913
Bohr model of the atom required van den Broek's atomic number of nuclear charges, and Bohr believed that Moseley's work contributed greatly to the acceptance of the model.
Atomic number
–
Henry Moseley helped develop the concept of atomic number by showing experimentally (1913) that Van den Broek's 1911 hypothesis combined with the
Bohr model nearly correctly predicted atomic X-ray emissions.
37.
Copper
–
Copper is a chemical element with symbol Cu and atomic number 29. It is a soft, malleable, and ductile metal with high thermal and electrical conductivity. A freshly exposed surface of copper has a reddish-orange color. Copper is one of the few metals that occur in nature in directly usable metallic form as opposed to needing extraction from an ore and this led to very early human use, from c.8000 BC. Copper used in buildings, usually for roofing, oxidizes to form a green verdigris, Copper is sometimes used in decorative art, both in its elemental metal form and in compounds as pigments. Copper compounds are used as agents, fungicides, and wood preservatives. Copper is essential to all living organisms as a trace dietary mineral because it is a key constituent of the enzyme complex cytochrome c oxidase. In molluscs and crustaceans, copper is a constituent of the blood pigment hemocyanin, replaced by the hemoglobin in fish. In humans, copper is found mainly in the liver, muscle, the adult body contains between 1.4 and 2.1 mg of copper per kilogram of body weight. The filled d-shells in these elements contribute little to interatomic interactions, unlike metals with incomplete d-shells, metallic bonds in copper are lacking a covalent character and are relatively weak. This observation explains the low hardness and high ductility of single crystals of copper, at the macroscopic scale, introduction of extended defects to the crystal lattice, such as grain boundaries, hinders flow of the material under applied stress, thereby increasing its hardness. For this reason, copper is supplied in a fine-grained polycrystalline form. The softness of copper partly explains its high conductivity and high thermal conductivity. The maximum permissible current density of copper in open air is approximately 3. 1×106 A/m2 of cross-sectional area, Copper is one of a few metallic elements with a natural color other than gray or silver. Pure copper is orange-red and acquires a reddish tarnish when exposed to air, as with other metals, if copper is put in contact with another metal, galvanic corrosion will occur. A green layer of verdigris can often be seen on old structures, such as the roofing of many older buildings. Copper tarnishes when exposed to sulfur compounds, with which it reacts to form various copper sulfides. There are 29 isotopes of copper, 63Cu and 65Cu are stable, with 63Cu comprising approximately 69% of naturally occurring copper, both have a spin of 3⁄2
Copper
–
Native copper (~4 cm in size)
Copper
–
A copper disc (99.95% pure) made by
continuous casting;
etched to reveal
crystallites.
Copper
–
Copper just above its melting point keeps its pink luster color when enough light outshines the orange
incandescence color.
Copper
–
Unoxidized copper wire (left) and oxidized copper wire (right).
38.
Messier object
–
The Messier objects are a set of over 100 astronomical objects first listed by French astronomer Charles Messier in 1771. The number of objects in the lists he published reached 103, a similar list had been published in 1654 by Giovanni Hodierna, but attracted attention only recently and was probably not known to Messier. The first edition covered 45 objects numbered M1 to M45, the first such addition came from Nicolas Camille Flammarion in 1921, who added Messier 104 after finding a note Messier made in a copy of the 1781 edition of the catalogue. M105 to M107 were added by Helen Sawyer Hogg in 1947, M108 and M109 by Owen Gingerich in 1960, M102 was observed by Méchain, who communicated his notes to Messier. Méchain later concluded that this object was simply a re-observation of M101, though sources suggest that the object Méchain observed was the galaxy NGC5866. Messiers final catalogue was included in the Connaissance des Temps for 1784 and these objects are still known by their Messier number from this list. Messier lived and did his work at the Hôtel de Cluny. The list he compiled contains only objects found in the sky area he could observe and he did not observe or list objects visible only from farther south, such as the Large and Small Magellanic Clouds. A summary of the astrophysics of each Messier object can be found in the Concise Catalog of Deep-sky Objects, in early spring, astronomers sometimes gather for Messier marathons, when all of the objects can be viewed over a single night
Messier object
–
All Messier objects. The pictures were taken and put together by an amateur astronomer
39.
Messier 29
–
Messier 29 is an open cluster in the Cygnus constellation. It was discovered by Charles Messier in 1764, and can be seen from Earth by using binoculars, the star cluster is situated in the highly crowded area of Milky Way near Gamma Cygni, at a distance of 7,200 or 4,000 light years. The Night Sky Observers Guide by Kepple and Sanner gives a value of 6,000 light years – the uncertainty due to poorly known absorption of the clusters light. According to the Sky Catalog 2000, Messier 29 is included in the Cygnus OB1 association and its age is estimated at 10 million years, as its five hottest stars are all giants of spectral class B0. The Night Sky Observers Guide gives the apparent brightness of the brightest star as 8.59 visual magnitudes, the absolute magnitude may be an impressive -8.2 mag, or a luminosity of 160,000 Suns. The linear diameter was estimated at only 11 light years and its Trumpler class is III,3, p, n, although Götz gives, differently, II,3, m, and Kepple/Sanner gives I,2, m, n. The Sky Catalogue 2000.0 lists it with 50 member stars and this cluster can be seen in binoculars. In telescopes, lowest powers are best, the brightest stars of Messier 29 form a stubby dipper, as Mallas says it. The four brightest stars form a quadrilateral, and another three, a north of them. It is often known as the tower due to its resemblance to the hyperboloid-shaped structures. A few fainter stars are around them, but the cluster appears quite isolated, in photographs, a large number of very faint Milky Way background stars shows up. Messier 29 can be quite easily as it is about 1.7 degrees south. In the vicinity of Messier 29, there is some diffuse nebulosity which can be detected in photographs, media related to Messier 29 at Wikimedia Commons Messier 29 LRGB image –2 hrs total exposure Messier 29, SEDS Messier pages Lawrence, Pete. Messier 29 on WikiSky, DSS2, SDSS, GALEX, IRAS, Hydrogen α, X-Ray, Astrophoto, Sky Map, Articles and images
Messier 29
–
M 29 open cluster, by Ole Nielsen
40.
Apparent magnitude
–
The apparent magnitude of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth. The brighter an object appears, the lower its magnitude value, the Sun, at apparent magnitude of −27, is the brightest object in the sky. It is adjusted to the value it would have in the absence of the atmosphere, furthermore, the magnitude scale is logarithmic, a difference of one in magnitude corresponds to a change in brightness by a factor of 5√100, or about 2.512. The measurement of apparent magnitudes or brightnesses of celestial objects is known as photometry, apparent magnitudes are used to quantify the brightness of sources at ultraviolet, visible, and infrared wavelengths. An apparent magnitude is measured in a specific passband corresponding to some photometric system such as the UBV system. In standard astronomical notation, an apparent magnitude in the V filter band would be denoted either as mV or often simply as V, the scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the sky were said to be of first magnitude, whereas the faintest were of sixth magnitude. Each grade of magnitude was considered twice the brightness of the following grade and this rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus. This implies that a star of magnitude m is 2.512 times as bright as a star of magnitude m +1 and this figure, the fifth root of 100, became known as Pogsons Ratio. The zero point of Pogsons scale was defined by assigning Polaris a magnitude of exactly 2. However, with the advent of infrared astronomy it was revealed that Vegas radiation includes an Infrared excess presumably due to a disk consisting of dust at warm temperatures. At shorter wavelengths, there is negligible emission from dust at these temperatures, however, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the scale was extrapolated to all wavelengths on the basis of the black body radiation curve for an ideal stellar surface at 11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance for the zero magnitude point, with the modern magnitude systems, brightness over a very wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30, astronomers have developed other photometric zeropoint systems as alternatives to the Vega system. The AB magnitude zeropoint is defined such that an objects AB, the dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor 5√100 ≈2.512. Inverting the above formula, a magnitude difference m1 − m2 = Δm implies a brightness factor of F2 F1 =100 Δ m 5 =100.4 Δ m ≈2.512 Δ m
Apparent magnitude
–
Asteroid
65 Cybele and two stars, with their magnitudes labeled
Apparent magnitude
–
30 Doradus image taken by
ESO 's
VISTA. This
nebula has an apparent magnitude of 8.
41.
Open cluster
–
An open cluster is a group of up to a few thousand stars that were formed from the same giant molecular cloud and have roughly the same age. More than 1,100 open clusters have been discovered within the Milky Way Galaxy and they are loosely bound by mutual gravitational attraction and become disrupted by close encounters with other clusters and clouds of gas as they orbit the galactic center. This can result in a migration to the body of the galaxy. Open clusters generally survive for a few hundred years, with the most massive ones surviving for a few billion years. In contrast, the more massive clusters of stars exert a stronger gravitational attraction on their members. Open clusters have been only in spiral and irregular galaxies. Young open clusters may not be contained within the cloud from which they formed. Over time, radiation pressure from the cluster will disperse the molecular cloud, typically, about 10% of the mass of a gas cloud will coalesce into stars before radiation pressure drives the rest of the gas away. Open clusters are key objects in the study of stellar evolution, because the cluster members are of similar age and chemical composition, their properties are more easily determined than they are for isolated stars. A number of clusters, such as the Pleiades, Hyades or the Alpha Persei Cluster are visible with the naked eye. Some others, such as the Double Cluster, are barely perceptible without instruments, the Wild Duck Cluster, M11, is an example. The prominent open cluster the Pleiades has been recognized as a group of stars since antiquity, while the Hyades forms part of Taurus, other open clusters were noted by early astronomers as unresolved fuzzy patches of light. The Roman astronomer Ptolemy mentions the Praesepe, the Double Cluster in Perseus, however, it would require the invention of the telescope to resolve these nebulae into their constituent stars. Indeed, in 1603 Johann Bayer gave three of these clusters designations as if they were single stars, the first person to use a telescope to observe the night sky and record his observations was the Italian scientist Galileo Galilei in 1609. When he turned the telescope toward some of the nebulous patches recorded by Ptolemy, he found they were not a single star, for Praesepe, he found more than 40 stars. Where previously observers had noted only 6-7 stars in the Pleiades, in his 1610 treatise Sidereus Nuncius, Galileo Galilei wrote, the galaxy is nothing else but a mass of innumerable stars planted together in clusters. Influenced by Galileos work, the Sicilian astronomer Giovanni Hodierna became possibly the first astronomer to use a telescope to find previously undiscovered open clusters, in 1654, he identified the objects now designated Messier 41, Messier 47, NGC2362 and NGC2451. Between 1774–1781, French astronomer Charles Messier published a catalogue of objects that had a nebulous appearance similar to comets
Open cluster
–
Star cluster NGC 3572 and its surroundings.
Open cluster
–
Mosaic of 30 open clusters discovered from
VISTA 's data. The open clusters were hidden by the dust in the Milky Way. Credit
ESO.
Open cluster
–
The colorful star cluster NGC 3590.
Open cluster
–
NGC 265, an open
star cluster in the
Small Magellanic Cloud
42.
Constellation
–
A constellation is formally defined as a region of the celestial sphere, with boundaries laid down by the International Astronomical Union. The constellation areas mostly had their origins in Western-traditional patterns of stars from which the constellations take their names, in 1922, the International Astronomical Union officially recognized the 88 modern constellations, which cover the entire sky. They began as the 48 classical Greek constellations laid down by Ptolemy in the Almagest, Constellations in the far southern sky are late 16th- and mid 18th-century constructions. 12 of the 88 constellations compose the zodiac signs, though the positions of the constellations only loosely match the dates assigned to them in astrology. The term constellation can refer to the stars within the boundaries of that constellation. Notable groupings of stars that do not form a constellation are called asterisms, when astronomers say something is “in” a given constellation they mean it is within those official boundaries. Any given point in a coordinate system can unambiguously be assigned to a single constellation. Many astronomical naming systems give the constellation in which an object is found along with a designation in order to convey a rough idea in which part of the sky it is located. For example, the Flamsteed designation for bright stars consists of a number, the word constellation seems to come from the Late Latin term cōnstellātiō, which can be translated as set of stars, and came into use in English during the 14th century. It also denotes 88 named groups of stars in the shape of stellar-patterns, the Ancient Greek word for constellation was ἄστρον. Colloquial usage does not draw a distinction between constellation in the sense of an asterism and constellation in the sense of an area surrounding an asterism. The modern system of constellations used in astronomy employs the latter concept, the term circumpolar constellation is used for any constellation that, from a particular latitude on Earth, never sets below the horizon. From the North Pole or South Pole, all constellations south or north of the equator are circumpolar constellations. In the equatorial or temperate latitudes, the term equatorial constellation has sometimes been used for constellations that lie to the opposite the circumpolar constellations. They generally include all constellations that intersect the celestial equator or part of the zodiac, usually the only thing the stars in a constellation have in common is that they appear near each other in the sky when viewed from the Earth. In galactic space, the stars of a constellation usually lie at a variety of distances, since stars also travel on their own orbits through the Milky Way, the star patterns of the constellations change slowly over time. After tens to hundreds of thousands of years, their familiar outlines will become unrecognisable, the terms chosen for the constellation themselves, together with the appearance of a constellation, may reveal where and when its constellation makers lived. The earliest direct evidence for the constellations comes from inscribed stones and it seems that the bulk of the Mesopotamian constellations were created within a relatively short interval from around 1300 to 1000 BC
Constellation
Constellation
Constellation
–
Babylonian tablet recording
Halley's comet in 164 BC.
Constellation
–
Chinese star map with a cylindrical projection (
Su Song)
43.
Cygnus (constellation)
–
Cygnus /ˈsɪɡnəs/ is a northern constellation lying on the plane of the Milky Way, deriving its name from the Latinized Greek word for swan. The swan is one of the most recognizable constellations of the summer and autumn. Cygnus was among the 48 constellations listed by the 2nd century astronomer Ptolemy, Cygnus contains Deneb, one of the brightest stars in the night sky and one corner of the Summer Triangle, as well as some notable X-ray sources and the giant stellar association of Cygnus OB2. One of the stars of this association, NML Cygni, is one of the largest stars currently known. The constellation is home to Cygnus X-1, a distant X-ray binary containing a supergiant. Many star systems in Cygnus have known planets as a result of the Kepler Mission observing one patch of the sky, in Greek mythology, Cygnus has been identified with several different legendary swans. The Greeks also associated this constellation with the story of Phaethon, the son of Helios the sun god. Phaethon, however, was unable to control the reins, forcing Zeus to destroy the chariot with a thunderbolt, according to the myth, Phaethons brother, Cycnus, grieved bitterly and spent many days diving into the river to collect Phaethons bones to give him a proper burial. The gods were so touched by Cycnuss devotion to his brother that they turned him into a swan, in Ovids Metamorphoses, there are three people named Cygnus, all of whom are transformed into swans. Alongside Cycnus, noted above, he mentions a boy from Tempe who commits suicide when Phyllius refuses to him a tamed bull that he demands. He also mentions a son of Neptune who is a warrior in the Trojan War who is eventually defeated by Achilles. In Polynesia, Cygnus was often recognized as a separate constellation, in Tonga it was called Tuula-lupe, and in the Tuamotus it was called Fanui-tai. Deneb was also given a name. The name Deneb comes from the Arabic name dhaneb, meaning tail, from the phrase Dhanab ad-Dajājah, in New Zealand it was called Mara-tea, in the Society Islands it was called Pirae-tea or Taurua-i-te-haapa-raa-manu, and in the Tuamotus it was called Fanui-raro. Beta Cygni was named in New Zealand, it was likely called Whetu-kaupo, Gamma Cygni was called Fanui-runga in the Tuamotus. The three-letter abbreviation for the constellation, as adopted by the IAU in 1922, is Cyg, the official constellation boundaries, as set by Eugène Delporte in 1930, are defined as a polygon of 28 segments. In the equatorial coordinate system, the right ascension coordinates of these borders lie between 19h 07. 3m and 22h 02. 3m, while the coordinates are between 27. 73° and 61. 36°. Covering 804 square degrees and around 1. 9% of the night sky, Cygnus culminates at midnight on 29 June, and is most visible in the evening from the early summer to mid-autumn in the Northern Hemisphere
Cygnus (constellation)
–
Cygnus as depicted in
Urania's Mirror, a set of constellation cards published in London c.1825. Surrounding it are Lacerta, Vulpecula and Lyra.
Cygnus (constellation)
–
List of stars in Cygnus
Cygnus (constellation)
–
The constellation Cygnus as it can be seen by the naked eye.
Cygnus (constellation)
–
V1331 Cyg is located in the dark cloud LDN 981.
44.
New General Catalogue
–
The NGC contains 7,840 objects, known as the NGC objects. It is one of the largest comprehensive catalogues, as it includes all types of space objects and is not confined to, for example. Dreyer also published two supplements to the NGC in 1895 and 1908, known as the Index Catalogues, describing a further 5,386 astronomical objects. Objects in the sky of the southern hemisphere are catalogued somewhat less thoroughly, the Revised New General Catalogue and Index Catalogue was compiled in 2009 by Wolfgang Steinicke. The original New General Catalogue was compiled during the 1880s by John Louis Emil Dreyer using observations from William Herschel and his son John, Dreyer had already published a supplement to Herschels General Catalogue of Nebulae and Clusters, containing about 1,000 new objects. In 1886, he suggested building a second supplement to the General Catalogue and this led to the publication of the New General Catalogue in the Memoirs of the Royal Astronomical Society in 1888. Assembling the NGC was a challenge, as Dreyer had to deal with many contradicting and unclear reports, while he did check some himself, the sheer number of objects meant Dreyer had to accept them as published by others for the purpose of his compilation. Dreyer was a careful transcriber and made few errors himself, and he was very thorough in his referencing, which allowed future astronomers to review the original references and publish corrections to the original NGC. The first major update to the NGC is the Index Catalogue of Nebulae and Clusters of Stars and it serves as a supplement to the NGC, and contains an additional 5,386 objects, collectively known as the IC objects. It summarizes the discoveries of galaxies, clusters and nebulae between 1888 and 1907, most of them made possible by photography, a list of corrections to the IC was published in 1912. The Revised New Catalogue of Nonstellar Astronomical Objects was compiled by Jack W. Sulentic and William G. Tifft in the early 1970s, and was published in 1973, as an update to the NGC. However, because the update had to be completed in just three summers, it failed to incorporate several previously-published corrections to the NGC data, and even introduced new errors. NGC2000.0 is a 1988 compilation of the NGC and IC made by Roger W. Sinnott and it incorporates several corrections and errata made by astronomers over the years. However, it too ignored the original publications and favoured modern corrections, the NGC/IC Project is a collaboration formed in 1993. It aims to identify all NGC and IC objects, and collect images, the Revised New General Catalogue and Index Catalogue is a compilation made by Wolfgang Steinicke in 2009. It is considered one of the most comprehensive and authoritative treatments of the NGC, messier object Catalogue of Nebulae and Clusters of Stars The Interactive NGC Catalog Online Adventures in Deep Space, Challenging Observing Projects for Amateur Astronomers
New General Catalogue
–
Spiral Galaxy NGC 3982 displays numerous spiral arms filled with bright stars, blue star clusters, and dark dust lanes. It spans about 30,000 light years, lies about 68 million light years from Earth and can be seen with a small telescope in the constellation of Ursa Major.
New General Catalogue
–
Four different
planetary nebulae. Clockwise starting from the top left:
NGC 6543,
NGC 7662,
NGC 6826, and
NGC 7009.
45.
Spiral galaxy
–
A spiral galaxy is a type of galaxy originally described by Edwin Hubble in his 1936 work The Realm of the Nebulae and, as such, forms part of the Hubble sequence. Spiral galaxies consist of a flat, rotating disk containing stars, gas and dust, and these are surrounded by a much fainter halo of stars, many of which reside in globular clusters. Spiral galaxies are named for the structures that extend from the center into the galactic disc. The spiral arms are sites of ongoing star formation and are brighter than the surrounding disc because of the young, hot OB stars that inhabit them. Roughly two-thirds of all spirals are observed to have a component in the form of a bar-like structure, extending from the central bulge. Our own Milky Way has recently confirmed to be a barred spiral. The most convincing evidence for its existence comes from a recent survey, performed by the Spitzer Space Telescope, together with irregular galaxies, spiral galaxies make up approximately 60% of galaxies in the local Universe. They are mostly found in low-density regions and are rare in the centers of galaxy clusters, Spiral arms are regions of stars that extend from the center of spiral and barred spiral galaxies. These long, thin regions resemble a spiral and thus give spiral galaxies their name, naturally, different classifications of spiral galaxies have distinct arm-structures. Sc and SBc galaxies, for instance, have very loose arms, whereas Sa, either way, spiral arms contain many young, blue stars, which make the arms so bright. A bulge is a huge, tightly packed group of stars, the term commonly refers to the central group of stars found in most spiral galaxies. Using the Hubble classification, the bulge of Sa galaxies is usually composed of Population II stars, further, the bulge of Sa and SBa galaxies tends to be large. In contrast, the bulges of Sc and SBc galaxies are much smaller and are composed of young, some bulges have similar properties to those of elliptical galaxies, others simply appear as higher density centers of disks, with properties similar to disk galaxies. Many bulges are thought to host a supermassive black hole at their centers, such black holes have never been directly observed, but many indirect proofs exist. In our own galaxy, for instance, the object called Sagittarius A* is believed to be a black hole. There is a correlation between the mass of the black hole and the velocity dispersion of the stars in the bulge. However, some stars inhabit a spheroidal halo or galactic spheroid, the orbital behaviour of these stars is disputed, but they may describe retrograde and/or highly inclined orbits, or not move in regular orbits at all. The galactic halo also contains many globular clusters, due to their irregular movement around the center of the galaxy—if they do so at all—these stars often display unusually high proper motion
Spiral galaxy
–
An example of a spiral galaxy, the
Pinwheel Galaxy (also known as Messier 101 or NGC 5457)
Spiral galaxy
–
Barred spiral galaxy UGC 12158.
Spiral galaxy
–
NGC 1300 in
infrared light.
Spiral galaxy
–
Spiral galaxy NGC 1345
46.
Andromeda (constellation)
–
Andromeda is one of the 48 constellations listed by the 2nd-century Greco-Roman astronomer Ptolemy and remains one of the 88 modern constellations. Located north of the equator, it is named for Andromeda, daughter of Cassiopeia, in the Greek myth. Andromeda is most prominent during autumn evenings in the Northern Hemisphere, because of its northern declination, Andromeda is visible only north of 40° south latitude, for observers farther south it lies below the horizon. It is one of the largest constellations, with an area of 722 square degrees. This is over 1,400 times the size of the moon, 55% of the size of the largest constellation, Hydra. Its brightest star, Alpha Andromedae, is a star that has also been counted as a part of Pegasus, while Gamma Andromedae is a colorful binary. Only marginally dimmer than Alpha, Beta Andromedae is a red giant, the constellations most obvious deep-sky object is the naked-eye Andromeda Galaxy, the closest spiral galaxy to the Milky Way and one of the brightest Messier objects. Several fainter galaxies, including M31s companions M110 and M32, as well as the more distant NGC891, the Blue Snowball Nebula, a planetary nebula, is visible in a telescope as a blue circular object. Andromeda is the location of the radiant for the Andromedids, a meteor shower that occurs in November. The uranography of Andromeda has its roots most firmly in the Greek tradition, the stars that make up Pisces and the middle portion of modern Andromeda formed a constellation representing a fertility goddess, sometimes named as Anunitum or the Lady of the Heavens. Andromeda is known as the Chained Lady or the Chained Woman in English and it was known as Mulier Catenata in Latin and al-Marat al Musalsalah in Arabic. Offended at her remark, the nymphs petitioned Poseidon to punish Cassiopeia for her insolence, Andromedas panicked father, Cepheus, was told by the Oracle of Ammon that the only way to save his kingdom was to sacrifice his daughter to Cetus. Perseus and Andromeda then married, the myth recounts that the couple had nine children together – seven sons, after Andromedas death Athena placed her in the sky as a constellation, to honor her. Several of the neighboring constellations also represent characters in the Perseus myth and it is connected with the constellation Pegasus. Andromeda was one of the original 48 constellations formulated by Ptolemy in his 2nd-century Almagest, in which it was defined as a specific pattern of stars. She is typically depicted with α Andromedae as her head, ο and λ Andromedae as her chains, and δ, π, μ, Β, however, there is no universal depiction of Andromeda and the stars used to represent her body, head, and chains. Arab astronomers were aware of Ptolemys constellations, but they included a second constellation representing a fish at Andromedas feet, several stars from Andromeda and most of the stars in Lacerta were combined in 1787 by German astronomer Johann Bode to form Frederici Honores. It was designed to honor King Frederick II of Prussia, in 1922, the IAU defined its recommended three-letter abbreviation, And
Andromeda (constellation)
–
Johannes Hevelius 's depiction of Andromeda, from the 1690 edition of his Uranographia. As was conventional for
celestial atlases of the time, the constellation is a mirror image of modern maps as it was drawn from a perspective outside the
celestial sphere.
Andromeda (constellation)
–
List of stars in Andromeda
Andromeda (constellation)
–
Andromeda as depicted in
Urania's Mirror, a set of constellation cards published in London c. 1825, showing the constellation from the inside of the celestial sphere
Andromeda (constellation)
–
Photo of the constellation Andromeda, as it appears to the naked eye. Lines have been added for clarity.
47.
Lunar calendar
–
A lunar calendar is a calendar based upon cycles of the Moons phases, in contrast to solar calendars based solely upon the solar year. A purely lunar calendar is also distinguished from lunisolar calendars whose lunar months are brought into alignment with the year through some process of intercalation. The details of when months begin varies from calendar to calendar, with using new, full, or crescent moons. Because each lunation is a less than 29 days,12 hours,44 minutes. Such holidays include Ramadan, the Chinese, Japanese, Korean, Vietnamese, and Mongolian New Year, the Nepali New Year, the Mid-Autumn Festival and Chuseok, Loi Krathong, and Diwali. The earliest known lunar calendar was found at Warren Field in Scotland and has dated to c. 8000 BC. Some scholars argue for lunar calendars still further back—Rappenglück in the marks on a c. 17, 000-year-old cave painting at Lascaux and Marshack in the marks on a c. 27, most calendars referred to as lunar calendars are in fact lunisolar calendars. Their months are based on observations of the cycle, with intercalation being used to bring them into general agreement with the solar year. The solar civic calendar of ancient Egypt showed traces of its origin in the lunar calendar. Present-day lunisolar calendars include the Chinese, Hindu, and Thai calendars, synodic months are 29 or 30 days in length, making a lunar year of 12 months about 11 days shorter than a solar year. Some lunar calendars do not use intercalation, such as most Islamic calendars, for those that do, such as the Hebrew calendar, the most common form of intercalation is to add an additional month every second or third year. Some lunisolar calendars are also calibrated by annual natural events which are affected by lunar cycles as well as the solar cycle, an example of this is the lunar calendar of the Banks Islands, which includes three months in which the edible palolo worm mass on the beaches. These events occur at the last quarter of the lunar month, lunar and lunisolar calendars differ as to which day is the first day of the month. In some lunisolar calendars, such as the Chinese calendar, the first day of a month is the day when a new moon occurs in a particular time zone. In others, such as some Hindu calendars, each month begins on the day after the moon or the new moon. Others were based in the past on the first sighting of a lunar crescent, the length of each lunar cycle varies slightly from the average value. In addition, observations are subject to uncertainty and weather conditions, thus to avoid uncertainty about the calendar, there have been attempts to create fixed arithmetical rules to determine the start of each calendar month. The average length of the month is 29.530589 days
Lunar calendar
–
Physical lunar features
48.
Saturn
–
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with a radius about nine times that of Earth. Although it has only one-eighth the average density of Earth, with its larger volume Saturn is just over 95 times more massive, Saturn is named after the Roman god of agriculture, its astronomical symbol represents the gods sickle. Saturns interior is composed of a core of iron–nickel and rock. This core is surrounded by a layer of metallic hydrogen, an intermediate layer of liquid hydrogen and liquid helium. Saturn has a yellow hue due to ammonia crystals in its upper atmosphere. Saturns magnetic field strength is around one-twentieth of Jupiters, the outer atmosphere is generally bland and lacking in contrast, although long-lived features can appear. Wind speeds on Saturn can reach 1,800 km/h, higher than on Jupiter, sixty-two moons are known to orbit Saturn, of which fifty-three are officially named. This does not include the hundreds of moonlets comprising the rings, Saturn is a gas giant because it is predominantly composed of hydrogen and helium. It lacks a definite surface, though it may have a solid core, Saturns rotation causes it to have the shape of an oblate spheroid, that is, it is flattened at the poles and bulges at its equator. Its equatorial and polar radii differ by almost 10%,60,268 km versus 54,364 km, Jupiter, Uranus, and Neptune, the other giant planets in the Solar System, are also oblate but to a lesser extent. Saturn is the planet of the Solar System that is less dense than water—about 30% less. Although Saturns core is considerably denser than water, the specific density of the planet is 0.69 g/cm3 due to the atmosphere. Jupiter has 318 times the Earths mass, while Saturn is 95 times the mass of the Earth, together, Jupiter and Saturn hold 92% of the total planetary mass in the Solar System. On 8 January 2015, NASA reported determining the center of the planet Saturn, the temperature, pressure, and density inside Saturn all rise steadily toward the core, which causes hydrogen to transition into a metal in the deeper layers. Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a rocky core surrounded by hydrogen. This core is similar in composition to the Earth, but more dense, in 2004, they estimated that the core must be 9–22 times the mass of the Earth, which corresponds to a diameter of about 25,000 km. This is surrounded by a liquid metallic hydrogen layer, followed by a liquid layer of helium-saturated molecular hydrogen that gradually transitions to a gas with increasing altitude
Saturn
–
Saturn in natural color, photographed by
Cassini in 2004
Saturn
Saturn
–
Composite image roughly comparing the sizes of Saturn and Earth
Saturn
–
Auroral lights at Saturn’s north pole.
49.
February
–
February is the second month of the year in the Julian and Gregorian calendars. It is the shortest month of the year as it is the month to have a length of less than 30 days. The month has 28 days in common years or 29 days in leap years, February is the third month of meteorological winter in the Northern Hemisphere. In the Southern Hemisphere, February is the last month of summer, February may be pronounced either as. Many people pronounce it as, as if it were spelled Feb-u-ary and this comes about by analogy with January, as well as by a dissimilation effect whereby having two rs close to each other causes one to change for ease of pronunciation. The Roman month Februarius was named after the Latin term februum, January and February were the last two months to be added to the Roman calendar, since the Romans originally considered winter a monthless period. They were added by Numa Pompilius about 713 BC, February remained the last month of the calendar year until the time of the decemvirs, when it became the second month. At certain intervals February was truncated to 23 or 24 days, the Straight Dope, How come February has only 28 days
February
–
February, from the
Très riches heures du Duc de Berry
February
–
February,
Leandro Bassano
February
–
Chocolates for
St. Valentine's Day
February
–
The
violet
50.
Leap year
–
A leap year is a calendar year containing one additional day added to keep the calendar year synchronized with the astronomical or seasonal year. By inserting an additional day or month into the year, the drift can be corrected, a year that is not a leap year is called a common year. For example, in the Gregorian calendar, each year has 366 days instead of the usual 365. In the Bahai Calendar, a day is added when needed to ensure that the following year begins on the vernal equinox. For example, Christmas Day fell on a Tuesday in 2001, Wednesday in 2002, the length of a day is also occasionally changed by the insertion of leap seconds into Coordinated Universal Time, owing to the variability of Earths rotational period. Unlike leap days, leap seconds are not introduced on a regular schedule, in the Gregorian calendar, the standard calendar in most of the world, most years that are multiples of 4 are leap years. In each leap year, the month of February has 29 days instead of 28, adding an extra day to the calendar every four years compensates for the fact that a period of 365 days is shorter than a tropical year by almost 6 hours. Some exceptions to this rule are required since the duration of a tropical year is slightly less than 365.25 days. For example, the years 1700,1800, and 1900 were not leap years, over a period of four centuries, the accumulated error of adding a leap day every four years amounts to about three extra days. The Gregorian calendar therefore removes three leap days every 400 years, which is the length of its leap cycle and this is done by removing February 29 in the three century years that cannot be exactly divided by 400. The years 1600,2000 and 2400 are leap years, while 1700,1800,1900,2100,2200 and 2300 are common years, by this rule, the average number of days per year is 365 + 1⁄4 − 1⁄100 + 1⁄400 =365.2425. The rule can be applied to years before the Gregorian reform, the Gregorian calendar was designed to keep the vernal equinox on or close to March 21, so that the date of Easter remains close to the vernal equinox. The Accuracy section of the Gregorian calendar article discusses how well the Gregorian calendar achieves this design goal, the following pseudocode determines whether a year is a leap year or a common year in the Gregorian calendar. The year variable being tested is the representing the number of the year in the Gregorian calendar. Care should be taken in translating mathematical integer divisibility into specific programming languages, if then else if then else if then else February 29 is a date that usually occurs every four years, and is called leap day. This day is added to the calendar in leap years as a corrective measure, the Gregorian calendar is a modification of the Julian calendar first used by the Romans. The Roman calendar originated as a calendar and named many of its days after the syzygies of the moon, the new moon. The Nonae or nones was not the first quarter moon but was exactly one nundina or Roman market week of nine days before the ides and this is what we would call a period of eight days
Leap year
–
A Swedish pocket calendar from the year 2008 showing February 29
Leap year
–
February 1900 calendar showing that 1900 was not a leap year
Leap year
–
In the older
Roman Missal, feast days falling on or after February 24 are celebrated one day later in leap year.
Leap year
–
A
spinster eagerly awaits the upcoming leap day, in this 1903 cartoon by
Bob Satterfield.